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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, Issue 4, Pages 53–63 (Mi vuu450)  

This article is cited in 6 scientific papers (total in 6 papers)

MATHEMATICS

On the property of uniform complete controllability of a discrete-time linear control system

V. A. Zaitsev, S. N. Popova, E. L. Tonkov

Department of Differential Equations, Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia
Full-text PDF (231 kB) Citations (6)
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Abstract: We study the property of uniform complete controllability (according to Kalman) for a discrete-time linear control system
\begin{equation} x(t+1)=A(t)x(t)+B(t)u(t),\quad t\in\mathbb N_0,\quad (x,u)\in\mathbb R^n\times\mathbb R^m. \end{equation}
We prove that if the system (1) is uniformly completely controllable, then the matrix $A(\cdot)$ is completely bounded on $\mathbb N_0$ (i.e. $\sup_{t\in\mathbb N_0}(|A(t)|+|A^{-1}(t)|)<+\infty$) and the matrix $B(\cdot)$ is bounded on $\mathbb N_0$. We prove that the system (1) is uniformly completely controllable if and only if there exists a $\vartheta\in\mathbb N$ such that for all $\tau\in\mathbb N_0$ the inequalities $\alpha_1I\leqslant W_1(\tau+\vartheta,\tau)\leqslant\beta_1I$, $\alpha_2I\leqslant W_2(\tau+\vartheta,\tau)\leqslant\beta_2I$ hold for some positive $\alpha_i$ and $\beta_i$, where
\begin{gather*} W_1(t,\tau)\doteq\sum_{s=\tau}^{t-1}X(t,s+1)B(s)B^*(s)X^*(t,s+1),\\ W_2(t,\tau)\doteq\sum_{s=\tau}^{t-1}X(\tau,s+1)B(s)B^*(s)X^*(\tau,s+1). \end{gather*}
On the basis of this statement, we prove the following criterion for uniform complete controllability of the system (1), which is similar to the Tonkov criterion of uniform complete controllability for continuous-time systems: the system (1) is $\vartheta$-uniformly completely controllable if and only if the matrix $A(\cdot)$ is completely bounded on $\mathbb N_0$; the matrix $B(\cdot)$ is bounded on $\mathbb N_0$; there exists an $\ell=\ell(\vartheta)>0$ such that for every $\tau\in\mathbb N_0$ and for any $x_1\in\mathbb R^n$ there exists a control function $u(t)$, $t\in[\tau,\tau+\vartheta)$, which transfers the solution of the system (1) from the state $x(\tau)=0$ to the state $x(\tau+\vartheta)=x_1$, and the inequality $|u(t)|\leqslant\ell|x_1|$ holds for all $t\in[\tau,\tau+\vartheta)$.
Keywords: linear control system, discrete time, uniform complete controllability.
Received: 15.08.2014
Document Type: Article
UDC: 517.977.1+517.929.2
MSC: 93B05, 93C05, 93C55
Language: Russian
Citation: V. A. Zaitsev, S. N. Popova, E. L. Tonkov, “On the property of uniform complete controllability of a discrete-time linear control system”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, no. 4, 53–63
Citation in format AMSBIB
\Bibitem{ZaiPopTon14}
\by V.~A.~Zaitsev, S.~N.~Popova, E.~L.~Tonkov
\paper On the property of uniform complete controllability of a discrete-time linear control system
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2014
\issue 4
\pages 53--63
\mathnet{http://mi.mathnet.ru/vuu450}
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  • This publication is cited in the following 6 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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